Spline Method for Nonlinear Optimal Thrust Vector Controls for Atmospheric Interceptor Guidance

Abstract

Theme T HE solution of a general rtth-order nonlinear system of differential equations is analytically approximated by an rtth-degree spline polynomial series. This spline approach is applied to determine the optimal thrust vector controls for guiding an interceptor to a target in dense atmosphere. The problem formulation utilizes the calculus of variations leading to a Two Point Boundary Value (TPBV) problem. The spline series, in conjunction with a oneor multidimensional parameter search procedure, provides a novel optimization technique for transforming the optimality state and adjoint nonlinear differential equations with specified boundary conditions into algebraic solution expressions that are easily calculated. Boundary conditions at the initial and terminal times are also satisfied algebraically on each sweep (iteration). A noniterative guidance law is obtained as an approximate solution to the TPBV problem. Two-dimensional atmospheric intercepts are considered with fixed terminal time (fixed altitude of intercept), zero terminal miss distance, and a quadratic performance index consist'*ng of the integral of the square of the thrust, which is a measure of fuel consumption. The quick convergence of the technique to the true optimal TPBV solution is illustrated with computer results under the severe conditions of a poor starting nominal, rapidly varying components of thrust, and large atmospheric drag nonlinearities.